Why Do We Need Foundations for Modelling Uncertainties?

1 What Are Foundations? Surely we want solid foundations. What kind of castle can we build on sand? What is the point of devoting effort to balconies and minarets, if the foundation may be so weak as to allow the structure to collapse of its own weight? We want our foundations set on bedrock, designed to last for generations. Who would want an architect who cannot certify the soundness of the foundations of his buildings? The architectural analogy is not entirely persuasive, though. It is easy to understand what the foundations of a building are: they are the steel pilings, the stone, the concrete, on which the rest of the edifice physically stands. It stands still. Science is not like that. It is living, dynamic, constantly changing. The foundations of a part of science are themselves open to change. So what are the foundations of a branch of science? There are the first principles, that everyone can and does (at least for the time being) agree on. The axioms of the discipline. Often these foundations are provided by a different branch of science. Thus engineering takes for granted-uses as a foundation-statics, kinematics, dynamics... In that sense the foundations of uncertainty clearly include logic and mathematics: we don't need to invent the real number system for ourselves. But there may be something more we want, something more domain specific, on which we can build our science. Some key idea, like the idea of 'force' in physics, or of a 'neighborhood' in topology. We have yet to agree on any such idea or set of ideas: we argue with each other and interrogate our intuitions. And disagree at length. We would liked to find relations, common elements, among the diverse ideas with which we work. But there is no unified theory, no general framework to which we can turn to see the relation between one idea and another. At the same time, though, we do manage to put together useful systems in which uncertainty figures. In fact people with intense and widely divergent views regarding the nature of uncertainty have been successful in making practical use of their ideas regarding uncertainty, its measurement, procedures for updating uncertainty, its role in decision making, etc. How is this possible? How can you do something right if you start off on the wrong foot? Clearly, since each of n people thinks the other n-1 are …